If you can replace it then even better. With the new one try to use wall charger only and perform a few full charge-discharge cycles as suggested on this forum as well. That may help you.
I did the "disconnect battery" trick as I tried to avoid sending the device for service from Europe to US. Apart of the fact that I didn't wanted to loose the reader for a month (sending there and back) I was worried that instead original reader I will receive refurbished one. So I did that trick because I somehow decided it is the only choice I have had
Anyway I'll be interested knowing if the replaced reader will behave better. From my perspective the reader battery is the weakest point in PRS-600 as very often people report issues related to this part. But once you make it work the tool becomes perfect